Modelamento e simulação de impacto balístico em sistema cerâmica-metal

UNIVERSIDADE FEDERAL DE SANTA CATARINA PROGRAMA DE PÓS GRADUAÇÃO EM CIÊNCIA E

ENGENHARIA DE MATERIAIS

Leandro Neckel

MODELAMENTO E SIMULAÇÃO DE IMPACTO BALÍSTICO EM SISTEMA CERÂMICA-METAL

Dissertação submetida ao Programa de Pós Graduação em Ciência e Engenharia de Materiais da Universidade Federal de Santa Catarina para obtenção do Grau de Mestre em Ciência e Engenharia de Materiais

Orientador: Prof. Hazim Ali Al-Qureshi, Ph.D.

Coorientador: Prof. Dachamir Hotza, Dr.-Ing.

Florianópolis 2012

Catalogação na fonte elaborada pela biblioteca da Universidade Federal de Santa Catarina

A ficha catalográfica é confeccionada pela Biblioteca Central. Tamanho: 7cm x 12cm

Fonte: Times New Roman 9.5 Maiores Informações em

Leandro Neckel

MODELAMENTO E SIMULAÇÃO DE IMPACTO BALÍSTICO EM SISTEMA CERÂMICA-METAL

Esta Dissertação foi julgada adequada para obtenção do Título de Mestre em Ciência e Engenharia de Materiais, e aprovada em sua forma final pelo Programa de Pós Graduação em Ciência e Engenharia de Materiais.

Florianópolis, 9 de Março de 2012

___________________________ Prof. Carlos Augusto Silva de Oliveira, Dr.

Banca Examinadora:

_______________________ Prof. Hazim Ali Al-Qureshi,

Ph.D,

Orientador, Presidente UFSC – CEM/EMC

_______________________ Prof. Oscar Rubem Klegues

Montedo, Dr. Eng, Membro Externo, UNESC

_______________________ Prof. Carlos Pérez Bergmann,

Dr. Ing,

Membro Externo, UFRGS – DEM

_______________________ Prof. Dachamir Hotza, Dr. Ing,

Coorientador UFSC – EQA

_______________________ Prof. Márcio Celso Fredel, Dr.

Ing, UFSC – EMC, Membro, UFSC

ACKNOWLEDGMENTS

Initially I would like to thank my master and advisor Prof. Hazim Ali Al-Qureshi, who made me “think like an engineer”, what is really hard for a person who holds a degree in Physics.

To the Universidade Federal de Santa Catarina and the Graduate Program of Materials Science and Engineering coordinators, professors and workers.

To my co-advisor, Prof. Dachamir Hotza, for the project support, as well as for the opportunity to spend a year in Hamburg.

To Dr. Rolf Janssen, my supervisor during my stay in Hamburg, for supporting my work in Hamburg.

To Dr. Acires Dias for the support to the Fracture and Damage Mechanics Conference in Croatia.

To the staff and fellow students during my time at TUHH: Manfred, Paula, Rodrigo, Hüssein, Wolfgang, Nils, Ezgi and Tobias. A special thanks for Anja, Bastian and Sascha, who helped me with every difficulty.

To my colleagues Murilo and João Gustavo for helping me with some engineering topics.

To my friends Alexandre, Adroaldo, Luis Paulo, Leonan and Mauricio who always pushed me forward when I was thinking of going back.

To my fiancée Camila with great affection who always encouraged me to never give up with her love and patience.

And to my family, for all the education, love and, specially, the insistence, because without you I would not be here.

Medicine makes people ill, mathematics make them sad and theology makes them sinful.

ABSTRACT

New ballistic protection systems have been created recently based on high-tech materials. One of the industry’s objectives is to develop lighter and stronger defensive systems, which allow higher mobility and greater safety for both vehicles and humans. This work studies mathematically the behavior of a protection against a projectile impact. The employed model, originally proposed by Al-Qureshi et al [1], includes a ceramic and metal layer system protection, and describes the projectile behavior and the impact absorption properties of the system. The literature also shows that the erosion tax and deceleration are highly dependent on the geometrical and structure parameters of the protective material. The phenomenon is described in different steps, presenting particular features for each step. The behavior equations present different system properties along the stages. This work presents a mathematical simulation performed on a developed model searching for best values for further studies. The properties to optimize include thicknesses of used plates, deforming profile, ceramic density, in addition to other relevant parameters. The program allowed the simulation of different parameters in the differential equations. As a result, graphs and surface plots were generated, which allowed a deeper analysis of the model and built an improved understanding of the process of fracture in materials by high velocity impact. Future studies will use these results as the basis for the manufacturing of a protection, which will be used for a practical experiment.

RESUMO

Novos sistemas de proteção balística vêm sendo recentemente criados com base em materiais de alta tecnologia. Um dos objetivos da indústria do ramo é desenvolver sistemas defensivos mais leves, porém mais fortes, que possibilitem ao portador, veicular ou humano, uma maior mobilidade com um maior nível de segurança. Este trabalho, baseado no trabalho original de Al-Qureshi et al [1], estuda matematicamente o comportamento de uma proteção contra um impacto de projétil. O modelo empregado inclui um sistema de proteção em camadas de cerâmica e metal, e ainda descreve o comportamento do projétil e as propriedades de absorção de impacto do sistema. A literatura ainda mostra que a taxa de erosão e desaceleração do projétil são altamente dependentes dos parâmetros geométricos e estruturais do material da proteção. O fenômeno de impacto e penetração é descrito em diferentes estágios, apresentando características particulares entre tais. As equações apresentadas demonstram diferentes propriedades do sistema ao longo dos estágios. Este trabalho ainda apresenta uma simulação matemática realizada sobre o modelo desenvolvido e aprimorado em busca de propriedades otimizadas do material para estudos futuros. Dentre as propriedades investigadas citam-se a espessura das placas utilizadas, o perfil de deformação do material metálico, a densidade da cerâmica, dentre outras características relevantes para o fenômeno. A rotina computacional possibilitou a aplicação de diferentes parâmetros nas equações propostas. Como resultado, gráficos e superfícies foram geradas, o

que possibilitou uma análise mais profunda do modelo e um maior entendimento do processo de fratura em materiais por impacto de alta velocidade. Estudos futuros utilizarão estes resultados e desenvolvimentos para a produção de uma proteção balística que será utilizada para um experimento prático.

FIGURE INDEX

Figure 2.1: Proposal of a two layer protection made by Al-Qureshi et al. [1] ... 24 Figure 2.2: First stage of the impact. The collision generates a shock wave that travels through the armor. ... 25 Figure 2.3: Projectile velocity as a function of penetration time.. 39 Figure 3.1: Evolution of the velocities of projectile and interface.57 Figure 4.1: Shape of the metallic plate after impact. ... 66 Figure 4.2 Deflection of the metal plate as a function of the radial distance with the original model equations. ... 67 Figure 5.1: evolution of the projectile velocity for different ceramic thicknesses. ... 76 Figure 5.2: Velocity(m/s) versus time(s) for the different parameters of mass, diameter and initial velocity of the projectiles. ... 79 Figure 5.3: Deformation w in function of the radial distance r and the hardening exponent n. ... 82 Figure 5.4: Deflection of the metallic plate w in function of the metal thickness h and radial distance r. ... 83

TABLE INDEX

Table 2.1: Projectile properties. ... 35 Table 2.2: Ceramic compositions. ... 35 Table 2.3: Summary of mechanical and physical properties of various compositions ... 37 Table 2.4: Comparison between theoretical and experimental values ... 38 Table 3.1: Values for the constants of the differential equation. . 51 Table 3.2: Basic initial parameters. ... 51 Table 3.3: Secondary initial conditions. ... 52 Table 3.4: Initial parameters for the second stage differential equations. ... 54 Table 3.5: Secondary initial parameters for the second stage differential equations. ... 54 Table 3.6: Deceleration, erosion and energy absorption rates among the stages. ... 59 Table 3.7: Values of the constants involved in equations 2.24 and 2.21. ... 60 Table 4.1: Vicker’s hardness and resistance against penetration in the ceramic for compositions B and C. ... 64 Table 4.2: Values of the constants used in the modified model. 71 Table 4.3: Comparison of the results between the original and modified theories of first stage duration and plastic deformation. ... 72 Table 5.1: Duration of the first stage for different ceramic thicknesses. ... 75

Table 5.2: Advance of the projectile in the ceramic tile for different considered plate thicknesses ... 77 Table 5.3: Parameters used in the simulation for different type of projectiles. ... 78 Table 5.4: Values of final deflection and penetration in the ceramic layer ... 80

SYMBOL LIST

A

Metal hardening law constant

p

A

Projectile effective area

c

Longitudinal velocity of the sound

d

Radius of the projectile

dV

Volume infinitesimal element

E

Elastic modulus

c

E

Elastic modulus of the ceramic material

k

E

Kinetic energy of the projectile

m

E

Elastic modulus of the metallic material

p

E

Plastic energy absorbed by the metallic plate

c

e

Thickness of the ceramic layer

m

e

Thickness of the metallic Plate

HV

Vicker’s Hardness

h

Thickness of the metal plate

k

Deflection profile constant

mod

k

Modified deflection profile constant )

(t

m Projectile mass function

pr

m

Remaining mass of the projectile after the second stage

n

Hardening index

c

R

Resistance against ceramic penetration

r

Radial distance

t Time

fs

t

Duration of the first stage )

(t

u Interface velocity function )

(t

v Projectile velocity function

pr

v

Projectile velocity in the end of the second stage )

(r

w Metal deflection profile function

0

w

Maximum deflection

p

Y

Dynamic yielding of the projectile material

Greek letters

ε

Average strain r

ε

Radial strain

ρ

Density c

ρ

Density of the ceramic material

m

ρ

Density of the metallic material

p

ρ

Density of the projectile material

σ

Average stress

TABLE OF CONTENTS 1 Introduction ... 21 1.1 Motivation ... 21 1.2 Objectives ... 22 2 Literature Review ... 23 2.1 Protection Modeling ... 23

2.2 The Geometry of the Protection ... 23

2.3 The Stages of Penetration ... 25

2.4 Equations of Movement ... 27

2.5 Equations for the Metallic Plate Deformation ... 30

2.6 Initial Results ... 34

3 Solving the Model ... 40

3.1 Equation for the Projectile Velocity in the First Stage ... 41

3.2 Equation for the Projectile Mass in the First Stage ... 43

3.3 Equation for the Projectile Velocity in the Second Stage .. 44

3.4 Equation for the Projectile Mass in the Second Stage .. 47

3.5 Duration of the First Stage ... 49

3.6 Solving the Movement and Mass Problems ... 50

3.7 Calculating the Metallic Deformation ... 59

4.1 Duration of the First Stage ... 61 4.2 The Resistance to the Penetration on Ceramic ... 62 4.3 Metallic Plate Deflection Profile ... 65 4.4 Comparison to the Modified Duration of First Stage ... 70 5 Simulations Based on the Modified Theories ... 73 5.1 Simulation on the Thicknesses of the Ceramic Layer .. 73 5.2 Simulation on Different Types of Projectiles ... 77 5.3 Simulation on the Metal Layer Properties ... 81 6 Conclusions ... 84 6.1 Concluding Remarks ... 84 6.2 Future Studies ... 85 7 References ... 87 8 Publication ... 90

21

1 INTRODUCTION

1.1 Motivation

Ballistic protections are structures that have been much studied by different areas of science and engineering. Those materials are often applied in the transport, security and construction sectors. The impact energy absorption and its effects are decisive on the issue of security and reliability to its application. In this way, better protection against impacts is a milestone for the development of new mobility systems. As the research in materials advances, faster and safer protective systems for vehicles and/or buildings can be built.

Al-Qureshi et al.[1] have developed a mathematical approach to the high speed impact against a particular type of protection and have performed some experiments with the proposed geometry as well. The armor cited is a two-layer system of ceramic and metal. The ceramic is responsible for the erosion of the incident projectile, which reduces its kinetic energy. The function of the metal layer is to prevent the projectile of going through the shield by absorbing its final kinetic energy as a plastic deformation.

It is important to cite that the high speed impacts are phenomena presenting high complexity and limited reproduction. Parameters such as pressure, energy and projectile incidence in the protection, for example, can hardly be adjusted and/or prevented. In the literature [2,3,4,5] there are several high

22 complexity models using different methods. Modern mathematical approaches, such the work of Iqbal et al. [2] and Vanichayangkuranont et al. [5], present the interpretation of the impact phenomenon based on finite element modeling. On the other hand, further models based on simpler approaches with different objectives have been developed. The work of Jerz et al. [3] and Naik et al. [4] present approaches based on the projectile energy and the plastic and elastic deformation theory.

However, the focus of the presented model is to represent the proposed impact situation using a simple approximation by ordinary differential equations.

1.2 Objectives

This work has the main objective of understanding the impact phenomenon on the proposed structure with its effects in both projectile and protection.

To achieve this goal, the following objectives were set • Solving the movement equations of the proposed

model using numerical methods;

• Generating a routine to solve the same problem with different parameters of both projectile and protection;

• Improving the model by modifying the proposed equations in their deficiencies;

• Simulating the impact over the proposed structures with different types of projectiles.

23

2 LITERATURE REVIEW

2.1 Protection Modeling

The process of impact is described by Al-Qureshi et al. [1] in three different stages, showing different behavior of the projectile and the protection for each one. The first one explains the initial impact, analyzing the erosion of the head of the projectile and the loss of velocity without effective penetration. The second stage shows the behavior of the ceramic in the energy absorption during the penetration. In this stage, a movement of an interface projectile-ceramic as one factor of deceleration is considered. The third step of the phenomenon reports the absorption of the remaining energy of the projectile. For the three stages there are equations representing the evolution of the velocity and mass of the projectile. Moreover, for the third stage it is presented a deterministic equation for the maximum deformation of the metal plate.

2.2 The Geometry of the Protection

Generally, impact problems were primarily of concern to the military, either for defensive or offensive purposes to develop armor or ammunition[1]. High speed impacts must be considered in many other situations. For all of them, it is important to prevent the complete failure of the protection by improving the mechanical properties of the used materials. Impacts of this nature are studied by many authors and each one researches the most

24 efficient protection for a specific projectile. For this particular geometry, the impact of a metallic projectile, more specifically a bullet, is considered.

Basically, the protection is a double layer system made of ceramic and metallic plates as shown in Figure 2.1. The ceramic layer cited in the literature is made of alumina and has a thickness between 8 and 12 mm. This plate is the one that will support the initial impact. The secondary layer is made of metallic material. The original work[1] proposes using stainless steel 304 with 10 mm thickness. This layer also may have a larger area than the ceramic plate. In fact, this is not necessary, but helpful in the construction of the protection. In addition, a layer of aramide ply lined with sikaflex 221 resin was applied over the ceramic plate in order to retain the projectile fragments after each test [1] This also helps to glue both layers.

Figure 2.1: Proposal of a two layer protection made by Al-Qureshi et al. [1]

25

2.3 The Stages of Penetration

The penetration process is divided in three stages which represent the behavior of both projectile and protection during the impact and penetration processes. These differences have to be considered to understand the system after the high speed impact. The first stage is related to the initial impact of the projectile in the ceramic. The impact generates a shock wave that travels through the shield and reflects back at the end of the protection. The generated wave is shown in Figure 2.2. The overlap of the incoming and reflected waves will crack the ceramic. This crack has a format of a cone due the wave overlapping process.

Figure 2.2: First stage of the impact. The collision generates a shock wave that travels through the armor.

An important factor that contributes in the first and second stage is the effect of the ceramic against the projectile. As known, the ceramic is brittle, but this material is incompressible after some pressure, which is very different from a metal. Once the projectile is made of metallic material, it is possible to prevent its

26 erosion in the high speed impact against the ceramic. In this case, the material of the first layer can be compared with powerful cutting tools with efficient machining capacity for the projectile material [6]. While the shock waves travel, the head of the projectile is eroded, but no effective penetration occurs.

The first stage ends when the wave returns to the generation point. Then, the wave overlapping ends and the ceramic is enough cracked to allow the projectile to penetrate effectively the first layer.

The second stage starts at same time. The process of effective penetration is analyzed in this stage. The projectile starts pushing an interface projectile-ceramic inwards the armor. This interface is responsible for the erosion in the projectile in this stage. It is important to observe that if there is no difference between the velocities of both projectile and interface, there will be no erosion. To better understand the process, it is necessary to consider the Tate’s law for solids subjected to extremely high pressures [7]. According to it, it is assumed that the rod (projectile) act as rigid body until a certain pressure is reached, which is a constant for a given material. At pressures above this value the material behaves hydrodynamically. A similar argument may be applied to the target material. In this case, however, the pressure required to make material flow hydrodynamically must overcome not only the rigidity of the material in the immediate neighborhood but also the inertia of surrounding material. In this way, Tate introduces the modified hydrodynamic law for solids,

27

which is necessary for the comprehension of penetration phenomenon.

This stage lasts until the velocities of the projectile and interface are equal. When this occurs, the erosion stops and the rest of the projectile keep penetrating and deforming the metal plate, which represents the final stage. The transition between the second and third stage is not exactly when the metal plate starts deforming, because it happens almost in the end of the intermediate stage. In addition, the deflection of the plate is studied with deterministic equations, while the velocity of the projectile is analyzed using a kinetic approach.

The third stage is the final deceleration of the projectile. In this situation, the interface projectile-ceramic has the same velocity of the bullet and both decelerations are equal. The metal layer of the protection deforms plastically absorbing the remaining energy of the bullet. Moreover, the protection will not fail completely if the ceramic has absorbed enough energy before the deflection starts.

2.4 Equations of Movement

As the penetration process, the equations are also divided by stages. Basically there are equations for the erosion and deceleration of the projectile for all the stages except the third where there is no erosion of the bullet.

To erode the head of the projectile then the dynamic yielding (Yp) of the projectile must be exceeded [1]. Then, the force opposing the penetration can be given as

28 p p

A

Y

t

v

dt

d

t

m

(

)

(

)

=

−

2.1 This Yp is one of the Tate’s [7] constants and represents

the maximum pressure that a solid supports before start behaving hydrodynamically. According to Tabor [8], the dynamic hardness of a metal is the pressure with which it resists local indentation by a rapidly moving indenter. The actual value of the dynamical yield pressure depends on the velocity of impact and on the way it is computed.

Ap is the projectile’s head area. m(t) and v(t) are the

function for mass and velocity of the projectile respectively. It is important to mention that mass is a function of time due the erosion of the bullet material. The erosion rate of the projectile is given by

)

(

)

(

t

A

v

t

m

dt

d

p p

ρ

−

=

2.2 where ρp is the density of the projectile material.

In this stage there is no movement of interface and the head of the bullet is being eroded without effective penetration. The shock wave generated by the impact travels through the protection and reflects back returning to the collision point. The respective necessary time was calculated by Wilkins [9,10] and it is given by

c

e

t

c fs

6

=

2.3 where ec is the thickness of the plate and c the longitudinal

29

When this time is reached, the rigidity of the material neighbor to the collision point is overcome and the bullet starts advancing through the ceramic. In this way, the movement of the interface projectile-ceramic begins. This fraction of the ceramic plate moves with velocity u(t) and is responsible for the erosion of the projectile during the second stage. In addition, in this intermediate stage, the force opposing the penetration remains the same.

The erosion rate in the second stage is given by

(

(

)

(

)

)

)

(

t

A

v

t

u

t

m

dt

d

p p

−

−

=

ρ

2.4 The configuration of the process during this stage can expressed by Tate’s hydrodynamic modified law [8]

(

)

2

( )

2

)

(

2

1

)

(

)

(

2

1

t

u

R

t

u

t

v

Y

_p

+

ρ

_p

−

=

_c

+

ρ

_c 2.5 where Rc is the dynamic resistance strength against penetration

into the ceramic [11], considered to be constant. This is the property related to the rigidity of the target proposed by Tate [7].

Simultaneously to the penetration of the projectile, the metallic base will move and will be deformed elastically. However, the energy produced by this elastic energy will be low and will be neglected [1]. The deformation of the metallic plate will be discussed in the next chapter.

As given by Eq. (2.4), the erosion will cease when both interface and projectile velocities are equal. Then, the second stage ends. After this occurs the projectile will continue penetrating the armor with deceleration given by

30 p c pr

v

t

R

A

dt

d

m

(

)

=

−

2.6 Here, mpr is the remaining mass of the projectile and due

the absence of erosion, and it is not a function of time anymore. 2.5 Equations for the Metallic Plate Deformation

Due the deceleration of the bullet caused by the ceramic layer, the metallic plate may not be perforated. In this case, the secondary layer will suffer plastic deformation and the plastic energy consumed by the plate can be expressed in terms of effective stress and strain as

dV

E

v p

∫ ∫

















∂

=

ε

σ

ε

0 2.7 For a material with effective stress–strain curve of a power-law hardening expression

( )

n

A

ε

σ

=

_2.8

Then, the total plastic energy becomes

( )

dV

n

A

E

v n p

∫

+

+

=

1

1

ε

_2.9.

Assuming that the material will be bulged in an axis symmetric mode, the effective strain can be written in terms of the radial strain as

r

ε

ε

=

2

_2.10

For a small displacement, the radial strain can be approximated to [1,12,13]

31 2

)

(

2

1













∂

∂

=

w

r

r

r

ε

2.11 where the function w(r) is the deflection profile of the impact. Thus

2

)

( 











∂

∂

=

w

r

r

ε

. 2.12

The geometry of the small dimple done by the impact can be approximated as a small cylinder. The element of volume can be expressed as

r

rh

dV

=

2

π

∂

_2.13

where h is the thickness of the plate. The substitution of Eqs. (2.12) and (2.13) into Eq. (2.9) gives

∫

∞ +

∂













∂

∂

+

=

0 ) 1 ( 2

)

(

1

2

r

r

r

w

r

n

hA

E

n p

π

. 2.14

For the solution of Eq. (2.14) is necessary to determine the deflection profile. Al-Qureshi et al. [14] have found that this profile can be expressed by an exponential given as

     ₋

=

dr k

e

w

r

w

(

)

₀ 2.15 where w0 is the maximum deflection of the dimple at r = 0,

and k is the deflection profile that can be determinate experimentally or by numerical adjustment. Moreover, it is necessary to consider the effect of the projectile geometry in the profile. Then, the constant d, which represents the bullet radius, was included in the equation. The differentiation of Eq. (2.15) gives

32       −













−

=

∂

∂

r d k

e

d

k

w

r

w

r

(

)

0 _{. 2.16}

that furnishes the deflection profile necessary in the Eq. (2.14). Substituting Eq. (2.16) into Eq. (2.14) gives

( ) ( )

r

r

e

d

kw

n

hA

E

d n kr n p



∂











−

+

=

_∫

∞     − + + 0 2 2 1 2 0

1

2

π

. 2.17

It is known that the strain hardening index n is fractional ( 0 < n < 1 ) and this will generate a complex value as result of

( )1 2 0 +













−

n

d

kw

.

Considering the physical nature of most constants it is possible to suppress the negative sign generated by the derivation demonstrated in Eq. (2.17). Also, the radial variation of the deflection profile is mirrored in the x-axis, which demonstrates that this ratio can be used as its own module. Now it gives

( ) ( )

r

r

e

d

kw

n

hA

E

d n kr n p



∂











+

=

_∫

∞     − + + 0 2 2 1 2 0

1

2

π

. 2.18

The solution for the expression is given by

3 ) 1 ( 2 0 2

)

1

(

2

1

+













=

+

n

w

d

k

hA

E

n n p

π

2.19 that can be written as

(

2( 1)

)

0 +

=

n p

B

w

E

2.20

33 where

( )

n

d

k

n

hA

B

2 3

1

2

1













+

=

π

. 2.21

According to Al-Qureshi et al. [1], a close examination of Eq. (2.16), reveals that the absorbed energy is directly proportional to the plate thickness, and is greatly influenced by the work hardening characteristics of the material.

After the initial impact, the movement of the interface starts deforming the metallic plate. It can be argued that, initially, the metal layer is compressed due to high pressure generated by the impact. In addition, the plate does not move significantly because of the low interface velocity. In this way, it is possible to affirm that the deflection of stainless steel plate starts at the end of the second stage of penetration.

Considering this, the energy absorbed by the plate can be approximately equated to the kinetic energy at the end of the second stage, which gives

p pr pr k

m

v

E

E

=

2

=

2

1

2.22 where vpr is the velocity of the projectile in the end of the second

stage. It gives

(

2( 1)

)

0 2

2

1

₊

=

n pr pr

v

B

w

m

2.23 and, then, the maximum deflection of the plate can be written as

34 ) 1 ( 2 1 2 0

2

1

+

















=

pr pr n

B

v

m

w

. 2.24 2.6 Initial Results

Al-Qureshi et al. [1] have performed some experiments based on the presented model. The target was attached to a special fixture to ensure a normal impact condition. Moreover, the experimental apparatus allowed measuring the initial velocity of the projectile in the collision. After each test, the maximum central deformation of the base (w0) and the residual mass of the

projectile (mpr) were measured. These tests permitted to compare

the theoretical and the experimental data. Then, these variables were used to determine the influence of the grain size on the efficiency of the shield.

The projectile properties are shown in Table 2.1. The same projectile caliber was used for all the experimental tests. The ceramic presented a thickness of 7.3, 9.3 and 11.3 mm. The ceramic plates were manufactured from the same chemical composition, but having different grain sizes. They were obtained by mixing different proportions of calcinated alumina A-1000SG and tubular alumina T-60 (both provided by Alcoa), and 2% TiO2 was added for the sintering operation. Table 2.2 lists the formulations used for those experiments [1].

The mechanical properties of the different ceramic compositions are listed in Table 2.3.

35

Table 2.1: Projectile properties.

Property Steel Nucleus

of the Projectile

Initial Velocity (m/s) 835

Mass (g) 9.54

Vicker’s Hardness (HV) 817.5

Dynamic Yield Stress (GPa) 2.82

Density (g/cm³) 8.41

Diameter (mm) 7.62

Table 2.2: Ceramic compositions. Composition Al2O3 A-1000SG (wt.%) Al2O3 Tubular T-60 (wt.%) TiO2 (wt.%) A 95 3 2 B 90 8 2 C 85 13 2 D 80 18 2

In addition, the stainless steel 304 had the following hardening power law

( )

0.29

MPa

935

ε

σ

=

_2.25

36 For comparison, the maximum deflection of the metal plate was calculated theoretically for all the different thicknesses of the ceramic plate. The examination of this values shows that the values predicted by the theory are generally in good agreement with experimental results. The difference between experimental and theoretical data can be attributed to several factors such as the fact that the projectile does not collide perpendicularly to the target. Also, the variation in the thickness of the ceramic and the density may affect the results. Other important factor is that the theory considers that the transverse sectional area of the projectile (Ap) remains constant during the

impact. This area is calculated based on the caliber of the projectile and then the theoretical fraction of mass lost in the impact was expected to be different of the experimental data. This comparison is shown in Table 2.4.

37

Table 2.3: Summary of mechanical and physical properties of various compositions Property Composition A B C D Modulus of Weibull (m) 14.4 8.4 8.8 15.1 σ50 (MPa) 203.7 175.0 171.3 161.3 σ0 (MPa) 209.0 182.8 178.5 165.2 Vicker’s Hardness (HV) - 1551.4 1259.8 - Rc (GPa) - 4.43 3.60 - Density (g/cm3) 3.79 3.90 3.80 3.75 Average Grain Size (µm) - 18±8 22±9 -

V50 is defined as the limit impact velocity for the

protection. It represents the velocity that causes partial or total penetration. From this investigation, it was found that the ceramic with composition C has an average V50 higher than the plate with

composition B. This evidence indicates that the grain size has an important influence on the absorber energy. Needless to say, more experimental results are needed before any final conclusion can be drawn with respect to the influence of grain size on the absorbing energy behavior [1]. Thus, it is important to consider that is necessary to have more experimental data before any final conclusion about the grain effect in the efficiency of the shield.

38 The analytical result for the evolution of both of the projectile and the interface velocities is shown in Figure 2.3. In this Figure, the projectile velocity v(t) is written as Vp(t) and the

interface velocity u(t) as Vi(t). The final deceleration (the third

stage) of the projectile is not demonstrated in this Figure. In this case, it is possible to observe that the interface velocity was zero during the first stage. On the other hand, the interface starts moving in the second stage until it reaches the same speed as the projectile. After this, as already mentioned, the erosion ceases and both the projectile and the interface move together with same velocity.

Table 2.4: Comparison between theoretical and experimental values Thickness of ceramic, ec (mm) Composition Impact Velocity, Vp (m/s) Maximum Deflection, w0 (mm) Exp. Theory Error (%) 11.3 B 792.7 16.5 17.0 -2.9 11.3 C 858.2 20.0 17.6 +13.6 9.3 B 628.9 18.0 17.8 +1.1 9.3 C 651.1 17.5 17.7 -1.1 7.3 B 428.8 15.5 17.3 -10.4 7.3 C 448.4 13.0 16.6 -21.7

39

The graph also shows that the second stage lasts more than the others revealing the importance of the ceramic layer in the energy absorption. In this case, it is possible to presume that different thicknesses of the ceramic layer will change the behavior of the shield in the impact situation. The use of thicker ceramic plates will increase the structure weight considerably and will reduce the deformation of the metal plate. On the other hand, ceramic with low thickness may not absorb enough energy of the projectile, which can cause total fracture of the armor.

40 3 SOLVING THE MODEL

Considering the present mathematical model, it is necessary to solve the equations to completely understand the behavior of the projectile and target in the situation of impact. Basically two different solutions can be found from the equations: one for the evolution of the mass and another for the velocity during the collision. Then, the solution for the interface velocity can be found from the kinematic equations system. On the other hand, the maximum deformation of the metallic plate after the impact can be calculated by using the deterministic equations of absorbed energy.

The division of the mathematical model in stages makes impossible to find a unique solution for the velocity or for the mass for the entire phenomenon. Considering this, the search for the solution of the impact evolution is also divided in stages, but then, all data are merged and a unique curve can be plotted. Then, for a more convenient and organized solution, the solutions are separated into mass and velocity equations for each stage, which makes necessary the manipulation and separation of the differential equations in a characteristic mathematical sentence for each situation.

After finding the equations of mass and velocity for each stage, it is possible to solve them by using numerical methods. In this way, initial conditions and constants values are required and can be easily found in the original work. At the shift of the first to the second stage the conditions of continuity are applied for both

41

mass and velocity. At the shift for the third stage, the continuity condition for the velocity is once more applied, but not for the mass since it is constant in this stage.

Since the kinematic equations are solved and the solutions are merged, it is possible to predict how much the metal plate will be deformed by applying values in the deterministic equation of deflection. The resulting values can be compared with some few results found in the original work. Moreover, the numeric values necessary for the constants involved in this equation can be found in Al-Qureshi’s work [1].

In the next chapters, the general method presented above will be explained in detail.

3.1 Equation for the Projectile Velocity in the First Stage

For the three stages, it is necessary to manipulate the equations to find a unique differential equation for the function of the mass and the velocity of the projectile. Needless to say, the function for the mass does not have to be solved for the third stage because it is constant.

The differential equation for the velocity on the first stage can be found firstly considering both equations of the model for the first stage. The force against the projectile is given by

p p

A

Y

t

v

dt

d

t

m

(

)

(

)

=

−

2.1 and the erosion tax is given by

)

(

)

(

t

A

v

t

m

dt

d

p p

ρ

−

=

. 2.2

42 First, the equation (2.1) can be derived in function of time giving

0

)

(

)

(

)

(

)

(

₂ 2

=













+

























t

v

dt

d

t

m

t

v

dt

d

t

m

dt

d

. 3.1

Now it is possible to substitute the erosion tax given by (2.2) in the equation (3.1). It gives

0

)

(

)

(

)

(

)

(

₂ 2

=













+













−

v

t

dt

d

t

m

t

v

dt

d

t

v

A

_p p

ρ

. 3.2

Also, it is possible to isolate the function m(t) in the equation (2.1), which is necessary for the equation (3.2). It gives













−

=

)

(

)

(

t

v

dt

d

A

Y

t

m

p p 3.3 Substituting equation (3.3) into (3.2) gives

0

)

(

)

(

)

(

)

(

2 2

=

























−













−

t

v

dt

d

t

v

dt

d

A

Y

t

v

dt

d

t

v

A

p p p p

ρ

3.4 that can be simplified and written as

p p

Y

t

v

dt

d

t

v

t

v

dt

d

2 2 2

(

)

(

)

)

(













−

=

ρ

3.5 The differential equation given by the expression (3.5) can be solved using numerical methods. The computational structure of the solving method will be explained in the next chapters.

43

It is interesting to mention that the velocity does not depend on the effective area of the projectile. On the other hand, it was expected that the higher the dynamic yielding Yp of the

projectile material, the lower the deceleration.

3.2 Equation for the Projectile Mass in the First Stage

To find the differential equation for the mass of the projectile in the first stage, it is again necessary to manipulate the expressions in function of the time for the first stage given by the equation (2.1) and (2.2). The derivation of equation (2.2) gives













−

=

(

)

)

(

2 2

t

v

dt

d

A

t

m

dt

d

p p

ρ

3.6 By isolating the acceleration term in the equation (2.1) it gives

)

(

)

(

t

m

A

Y

t

v

dt

d

₌

₋

p p 3.7 and this expression can substituted in the equation (3.6). Then the expression for the mass of the projectile in the first stage can be written as

)

(

)

(

2 2 2

t

m

Y

A

t

m

dt

d

₌

ρ

p p p 3.8 Here the importance of the geometry of the material can be noticed. The effective area, which is calculated using the diameter of the projectile, has a very high influence in the erosion tax. It is possible to note that the erosion tax will become lower

44 with the time. This can be explained by the drastic decrease of kinetic energy due to reduction of speed and mass.

In addition, it is known that the variation of the mass during the impact implies a variation in the projectile geometry. However, in this work the projectile area (Ap) during the whole

phenomenon is considered constant.

3.3 Equation for the Projectile Velocity in the Second Stage The second stage starts when the wave generated by the impact reflects back in the end of the protection and returns to the original point. After this, the movement starts being governed by different equations due to the existence of the interface projectile-ceramic movement. Again, to find the characteristic differential equation for the velocity in the second stage of penetration it is necessary to manipulate the equations given by Al-Qureshi’s model.

The force against the projectile remains the same as the first stage and it is given by

p p

A

Y

t

v

dt

d

t

m

(

)

(

)

=

−

2.1

and the erosion tax is now given by

(

(

)

(

)

)

)

(

t

A

v

t

u

t

m

dt

d

p p

−

−

=

ρ

2.4

where u(t) represents the interface velocity.

Then, for dealing with the problem of the velocity and mass for the second stage, it is necessary to use the Tate’s hydrodynamic modified law for solids given by

45

(

)

2

( )

2

)

(

2

1

)

(

)

(

2

1

t

u

R

t

u

t

v

Y

_p

+

ρ

_p

−

=

_c

+

ρ

_c 2.5

where Rc is the resistance of the ceramic against the penetration

and ρc is the ceramic density.

From the equation (2.4) is possible to obtain

)

(

)

(

)

(

v

t

A

t

m

dt

d

t

u

p p

+

=

ρ

_3.9

and from equation (3.1), which is also valid for the second stage, is possible to isolate the erosion tax, which gives

























−

=

)

(

)

(

)

(

)

(

2 2

t

v

dt

d

t

v

dt

d

t

m

t

m

dt

d

3.10 The replacement of equation (3.10) into (3.9) gives

)

(

)

(

)

(

)

(

)

(

2 2

t

v

A

t

v

dt

d

t

v

dt

d

t

m

t

u

p p

+

























−

=

ρ

3.11 and can be simplified using equation (3.3), which gives

)

(

)

(

)

(

)

(

₂ 2 2

t

v

t

v

dt

d

t

v

dt

d

Y

t

u

p p

+

























=

ρ

3.12

46 Tate’s equation (2.5) presents only the functions for the projectile and interface velocities and also can be manipulated to be written as a second degree equation for the function u(t):

(

)

2

(

)

(

)

2

2

(

)

0

)

(

t

2

−

+

v

t

u

t

+

R

−

Y

−

v

t

2

=

u

ρ

_c

ρ

_p

ρ

_{p c p}

ρ

_p

3.13 where, in the ax²+bx+c=0 format, the constants a, b and c are

(

)

2

)

(

2

2

;

)

(

2

;

b

v

t

c

R

Y

v

t

a

=

ρ

_c

−

ρ

_p

=

ρ

_p

=

_c

−

_p

−

ρ

_p 3.14 The solution for the equation (3.13) is given by

c p p p c p p c c c p c p

v

t

v

t

R

Y

R

Y

t

u

ρ

ρ

ρ

ρ

ρ

ρ

ρ

ρ

ρ

−

−

+

+

−

−

=

(

)

(

)

2

2

2

2

)

(

2 3.15 Now, substituting equation (3.15) into (3.12) is obtained

c p p p c p p c c c p c p p p _{vt vt R Y R Y} t v t v dt d t v dt d Y ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ − − + + − − = +             2 2 2 2 ) ( ) ( ) ( ) ( ) ( 2 2 2 2 3.16 that can be simplified finding the characteristic differential equation for the velocity on the second stage:

(

)

      − + + − −       = c p p p p c p p c c c p c p p Y Y R Y R t v t v t v dt d t v dt d ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ () () () 2 2 2 2 ) ( 2 2 2 2 3.17 The positive signal in the solution of the second degree equation was suppressed due to the possible complex solutions. Since the phenomenon is purely physical and real, it makes

47

absolutely no sense to consider imaginary solutions for the problem.

3.4 Equation for the Projectile Mass in the Second Stage The same manipulation has to be done to find the differential equation for the erosion of the projectile in the second stage. Initially, it is possible to substitute equation (3.9) into (3.15), which gives c p p p c p p c c c p c p p p Y R Y R t v t v t v A t m dt d ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ − − + + − − = + () () () 2 2 2 2 ) ( 2 3.18 that needs to be manipulated by isolating the function of velocity, which gives p p c p c p c p c p p p c c A t m dt d A R A Y t m dt d t v

ρ

ρ

ρ

ρ

ρ

ρ

ρ

ρ

ρ

_{() 2} 2 2 ₂ 2 2 ₍₎ 2 ) (       + − +       − = 3.19 This equation can be derived in function of t giving

p p c p c p c p c p p p c p c c A t m dt d A R A Y t m dt d t m dt d t m dt d t v dt d ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ₂ 2 2 2 2 2 2 2 2 ) ( 2 2 ) ( ) ( ) ( ) (       + −             +       − = 3.20 Here it is possible to use the equation (3.7) once more. Then it gives

48 p p c p c p c p c p p p c p c c p p A t m dt d A R A Y t m dt d t m dt d t m dt d t m A Y ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ 2 2 2 2 2 2 2 2 2 ) ( 2 2 ) ( ) ( ) ( ) (       + −             +       − = − 3.21 that can be written with the second derivative of the mass function isolated, which gives

                    + −       + − − = 2 2 2 2 2 2 2 2 ) ( 2 2 ) ( ) ( ) ( t m dt d A R A Y t m dt d t m Y A t m dt d p c p c p c p p p c p c c p p p c

ρ

ρ

ρ

ρ

ρ

ρ

ρ

ρ

ρ

ρ

ρ

3.22 The expression above is the characteristic differential equation for the loss of the mass in the second stage of penetration. Needless to say, both mass and velocity evolutions in the intermediary stage are strongly influenced by the projectile and ceramic materials properties. Then, at the end of this stage a small compression in the metal layer is initiated. However, due to the high speed and kinetic energy of the projectile, the effect of the resistance to the plastic deformation of the stainless steel plate was neglected.

The second stage ends when the erosion ceases. In this situation both interface and projectile velocities are equal and the loss of mass flux becomes zero. This is easily observable in equation (2.4). Considering this, the constant mass can be represented as mpr in the third stage equation given by

49 p c pr

v

t

R

A

dt

d

m

(

)

=

−

2.6

that can be solved with simple analytical methods. However, respecting the chosen methodology, the numerical method was once more chosen to find the solution. The method will be demonstrated in the next chapters.

3.5 Duration of the First Stage

To start solving the differential equations of the first stage, it is necessary to know how much time this initial part of the phenomenon will last. Considering the equation (3), it is possible to calculate this time due to the geometric parameters of the protection. The time of the first stage is given by

c

e

t

c fs

6

=

2.3 where ec is the thickness of the ceramic layer and c the

longitudinal velocity of sound in the material. For plates having 10 mm thickness, the time to form the fracture cone is approximately equal to 6 µs [1], which was used for the initial calculations.

In the end of the first stage, it is necessary to apply continuity conditions to the problem. It means that the final values for mass and velocity and their rates must be equal at the end of the first stage and at the beginning of the second stage. The same occurs at the end of the second stage with the velocity. It is necessary to mention that the influence of the epoxy resin between the ceramic and metal layer was not considered in the solution since it was also not take in account in the original model.

50 The erosion rate in the third stage is zero and the continuity condition does not need to be applied in this case. Moreover, the numerical method for solution and the storage method for the numerical values of the solution allow the easier manipulation of the data. These methods will be explained in the next chapters. 3.6 Solving the Movement and Mass Problems

The software Maplesoft Maple V12 was used to solve the equations and also to manipulate the data generated by the process of numerical solution. Firstly it was necessary to declare the characteristic differential equations for mass and velocity for the first stage. Then, it was also necessary to input the initial conditions for the problem. The second degree ordinary differential equation need two initial conditions: the primary condition is a value of the function in a specific value of abscise, and the secondary condition is a value of the first derivative in the same value of abscise of the primary condition. Needless to say that was also necessary to declare the numerical value for all the constants in the equations.

All the required constants values can be found in Al-Qureshi’s [1] work and are presented in Table 3.1.

51

Table 3.1: Values for the constants of the differential equation.

Constant Value Yp 2.82 GPa Ap 45.58 mm² ρp 8.41 g/cm³ Rc 4.43 GPa ρc 3.90 g/cm³

The dynamic yielding Yp of the projectile material is

considered constant during all collision. Moreover, the value of Ap

was calculated using the diameter of the projectile (7.62 mm) and it was also considered constant during all impact. The values of Rc and ρc are from the composition B (90% Al2O3 A-1000SG , 8% Al2O2 tubular T-60 , 2% TiO2 ) which had an average grain size of 18.4 µm.

The primary initial conditions for the ordinary differential equations can be also find in the original work and are shown in Table 3.2

Table 3.2: Basic initial parameters.

Input Value

Initial velocity : v(t=0) 835.0 m/s

52 Considering the second degree differential equations, it is also necessary to introduce the secondary initial conditions. Those conditions can be deduced from the original equations. Table 3.3 presents those conditions.

The process of computing also must be stopped at the correct time, otherwise the solution will be obtained for times longer than the duration of the first stage. For the same stage the software was programmed to generate sixty values for the evolutions of mass and velocity. With these values, it is already possible to plot the first stage curves of mass and velocity.

To the second stage, it is necessary to apply continuity conditions. The final values in the first stage have to be considered the initial parameters for the second stage equations. Then, considering the second order ordinary differential equations, it is necessary to input the final values of the derivative of mass and velocity in the first stage as initial parameters for the second stage too.

Table 3.3: Secondary initial conditions.

Input Value Initial acceleration

)

0

(

)

0

(

=

−

=

=

t

v

A

Y

t

v

dt

d

p p

-1.34*10

7

m/s²

Initial mass rate

)

0

(

)

0

(

t

=

=

−

A

v

t

=

m

dt

d

p p

ρ

-320.08

kg/s

53

The values of mass and velocity in the end of the first stage can be picked from the numerical solutions. Otherwise, the values of the derivatives must be calculated using the main equations of the second stage of penetration. The equations are

)

(

)

(

t

m

A

Y

t

v

dt

d

₌

₋

p p 3.7 and

(

(

)

(

)

)

)

(

t

A

v

t

u

t

m

dt

d

p p

−

−

=

ρ

. 2.4

In the second stage the interface projectile-ceramic starts to move with velocity u(t). This velocity is null in the beginning of this stage and can be considered this way in equation 2.4 to find the secondary initial parameters. The final results for mass and velocity in the first stage are demonstrated in Table 3.4 and the secondary initial parameters are shown in Table 3.5.

With the equations and initial conditions declared, it is now possible to program the software to solve the second stage equations. The stop criteria for the calculations must be the value of time when the velocity of projectile is the same as the speed of the interface projectile-ceramic. For this, it is necessary to compare both numerical solutions. Unfortunately, the solution for the interface velocity depends on the solutions for velocity and erosion rate, as shown in equation (3.9).